IJSTE - International Journal of Science Technology & Engineering | Volume 2 | Issue 12 | June 2016 ISSN (online): 2349-784X All rights reserved by www.ijste.org 140 Generalization of Single-Valued Mapping with Contraction Concept in Complex-Valued Metric Space Nisha Sharma Mamta Rani Department of Mathematics Department of Mathematics Manav Rachna International University Faridabad Pt. JLN Govt. College Faridabad Abstract Owning the concept of complex-valued metric spaces, which was introduced by Azam et al. [4], who introduced the new concept and established a common fixed point result in the context of complex-valued metric spaces. In this paper, we generalized the concept of chatterjea [14] contraction mapping for single-valued mapping on the complex-valued metric spaces. Keywords: Complex-valued metric space, single-valued mappings, Contraction mappings, Common fixed point, lower bound and greater lower bound (g.l.b) MSC: 46S40, 47H10, 54H25 ________________________________________________________________________________________________________ I. INTRODUCTION It is well known fact that the mathematical results regarding fixed points of contraction-type mappings are very useful for determining the existence and uniqueness of solutions to various mathematical models. The theory of fixed points has been developed, regarding the results to finding the fixed points self and nonself over the last 51 years. Many authors have proved fixed point results in the different kind of generalization in complex-valued metric spaces. Nadler [9] and Markin [8] was initiated the study of fixed points for multi-valued contraction mappings. Azam et al. [4] introduced the concept of complex-valued metric space and obtained sufficient conditions for the existence of common fixed points. Very recently, Ahmad et al. [2] obtained some new fixed point results for multi-valued mappings in the setting of complex-valued metric spaces. Some fixed point results by generalizing the contractive conditions in the context of complex-valued metric spaces was established by Sitthikul and Saejung [13] and Klin-eam and Suanoom [7]. The results presented in this paper substantially extend the results given by chatterjea et.al [14] for the multi-valued mappings. II. PRELIMINARIES Let C be the set of complex numbers and z1 ,z2  C. Define a partial order ≾ C on as follows: z1 ≾ z2 if and only if Re(z1) ≤ Re(z2), Im(z1) ≤ Im(z2). It follows that z1 ≾ z2 if one of the following conditions is satisfied: 1) Re(z1) = Re(z2), Im(z1) < Im(z2), 2) Re(z1) < Re(z2), Im(z1) = Im(z2), 3) Re(z1) < Re(z2), Im(z1) < Im(z2), 4) Re(z1) = Re(z2), Im(z1) = Im(z2), In particular, we will write z1⋦ z2 if z1 ≠ z2 and one of (i),(ii) and (iii) is satisfied and we will write z 1 ≺ z2 if only (iii) is satisfied. Note that 0 ≾ z1 ⋦ z2 ⟹ z1 < z2 , z1 ≾ z2 , z2 ≺ z3 ⟹ z1 ≺ z3 . Definition 1.3 Let X is a nonempty set. Suppose that the mapping d : X ⨯ X → C satisfies: 1) 0 ≾ d(x,y) for all x,y ∈ X and d(x,y) = 0 if and only if x = y; 2) d(x,y) = d(y,x) for all x,y ∈ X 3) d(x,y) ≾ d(y,x) + d(z,y) for all x,y,z ∈ X. Then d is called a complex-valued metric on X, and (X,d) is called a complex-valued metric space. A point x ∈ X is called an interior point of a set A ⊆ X whenever there exists 0 ≺ r ∈ C such that B(x,r) = {y ∈ X : d (x,y) ≺ r} ⊆ A.